A Finitely Stable Edit Distance for Functions Defined on Merge Trees

TL;DR

This paper introduces a new stable metric for comparing functions on merge trees, enabling effective topological data analysis in scenarios where traditional tools like persistence diagrams are inadequate.

Contribution

It defines a finitely stable edit distance for functions on merge trees, with stability properties and a computational approach, expanding TDA capabilities.

Findings

Effective comparison of functions on merge trees demonstrated with simulated data

The metric shows stability properties within TDA framework

Outperforms traditional methods like persistence diagrams in certain cases

Abstract

In this work we define a metric structure to compare functions defined on different merge trees. The metric introduced possesses some stability properties, which we illustrate within a standard topological data analysis (TDA) framework, and can be computed with a dynamical binary linear programming approach. We showcase the effectiveness of the whole framework with simulated data sets. Using functions defined on merge trees proves to be very effective in situations where other topological data analysis tools, like persistence diagrams, cannot be used meaningfully.